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sage:from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(12992, base_ring=CyclotomicField(336))
M = H._module
chi = DirichletCharacter(H, M([168,21,56,276]))
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gp:[g,chi] = znchar(Mod(10363, 12992))
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magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("12992.10363");
| Modulus: | \(12992\) |
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sage:chi.modulus()
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gp:g[1][1]
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magma:Modulus(chi);
|
| Conductor: | \(12992\) |
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sage:chi.conductor()
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gp:znconreyconductor(g,chi)
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magma:Conductor(chi);
|
| Order: | \(336\) |
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sage:chi.multiplicative_order()
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gp:charorder(g,chi)
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magma:Order(chi);
|
| Real: | no |
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sage:chi.multiplicative_order() <= 2
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gp:charorder(g,chi) <= 2
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magma:Order(chi) le 2;
|
| Primitive: | yes |
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sage:chi.is_primitive()
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gp:#znconreyconductor(g,chi)==1
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magma:IsPrimitive(chi);
|
| Minimal: | yes |
| Parity: | odd |
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sage:chi.is_odd()
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gp:zncharisodd(g,chi)
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magma:IsOdd(chi);
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\(\chi_{12992}(3,\cdot)\)
\(\chi_{12992}(19,\cdot)\)
\(\chi_{12992}(171,\cdot)\)
\(\chi_{12992}(243,\cdot)\)
\(\chi_{12992}(395,\cdot)\)
\(\chi_{12992}(467,\cdot)\)
\(\chi_{12992}(619,\cdot)\)
\(\chi_{12992}(635,\cdot)\)
\(\chi_{12992}(859,\cdot)\)
\(\chi_{12992}(955,\cdot)\)
\(\chi_{12992}(1083,\cdot)\)
\(\chi_{12992}(1123,\cdot)\)
\(\chi_{12992}(1291,\cdot)\)
\(\chi_{12992}(1419,\cdot)\)
\(\chi_{12992}(1587,\cdot)\)
\(\chi_{12992}(1755,\cdot)\)
\(\chi_{12992}(2131,\cdot)\)
\(\chi_{12992}(2299,\cdot)\)
\(\chi_{12992}(2467,\cdot)\)
\(\chi_{12992}(2595,\cdot)\)
\(\chi_{12992}(2763,\cdot)\)
\(\chi_{12992}(2803,\cdot)\)
\(\chi_{12992}(2931,\cdot)\)
\(\chi_{12992}(3027,\cdot)\)
\(\chi_{12992}(3251,\cdot)\)
\(\chi_{12992}(3267,\cdot)\)
\(\chi_{12992}(3419,\cdot)\)
\(\chi_{12992}(3491,\cdot)\)
\(\chi_{12992}(3643,\cdot)\)
\(\chi_{12992}(3715,\cdot)\)
...
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sage:chi.galois_orbit()
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gp:order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
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magma:order := Order(chi);
{ chi^k : k in [1..order-1] | GCD(k,order) eq 1 };
\((11775,2437,3713,4033)\) → \((-1,e\left(\frac{1}{16}\right),e\left(\frac{1}{6}\right),e\left(\frac{23}{28}\right))\)
| \(a\) |
\(-1\) | \(1\) | \(3\) | \(5\) | \(9\) | \(11\) | \(13\) | \(15\) | \(17\) | \(19\) | \(23\) | \(25\) |
| \( \chi_{ 12992 }(10363, a) \) |
\(-1\) | \(1\) | \(e\left(\frac{323}{336}\right)\) | \(e\left(\frac{325}{336}\right)\) | \(e\left(\frac{155}{168}\right)\) | \(e\left(\frac{5}{336}\right)\) | \(e\left(\frac{25}{112}\right)\) | \(e\left(\frac{13}{14}\right)\) | \(e\left(\frac{1}{6}\right)\) | \(e\left(\frac{55}{336}\right)\) | \(e\left(\frac{23}{168}\right)\) | \(e\left(\frac{157}{168}\right)\) |
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sage:chi(x)
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gp:chareval(g,chi,x) \\\\ x integer, value in Q/Z'
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magma:chi(x)
